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hep-th/0203213ITEP-TH-19/02HUTP-02/A004

Rational Conformal Field Theories andComplex Multiplication

Sergei Gukov and Cumrun Vafa

Jefferson Physical Laboratory

Harvard University

Cambridge, MA 02138, USA

We study the geometric interpretation of two dimensional rational conformal field

theories, corresponding to sigma models on Calabi-Yau manifolds. We perform a detailed

study of RCFT’s corresponding to T 2 target and identify the Cardy branes with geometric

branes. The T 2’s leading to RCFT’s admit “complex multiplication” which characterizes

Cardy branes as specific D0-branes. We propose a condition for the conformal sigma model

to be RCFT for arbitrary Calabi-Yau n-folds, which agrees with the known cases. Together

with recent conjectures by mathematicians it appears that rational conformal theories are

not dense in the space of all conformal theories, and sometimes appear to be finite in

number for Calabi-Yau n-folds for n > 2. RCFT’s on K3 may be dense. We speculate

about the meaning of these special points in the moduli spaces of Calabi-Yau n-folds in

connection with freezing geometric moduli.

March 2002

1. Introduction

Two dimensional “Rational Conformal Field Theories” (RCFT) were introduced in

[1] as a particularly nice class of conformal field theories which have more structure and

could be potentially classified. Moreover it was suggested that perhaps they may be

dense in the space of all conformal theories, and so in this way one can potentially get a

handle on all conformal theories. RCFT’s are characterized by having a symmetry algebra

extending the Virasoro algebra (chiral algebra), in terms of which the Hilbert space can

be decomposed into finite irreducible representations. Thus RCFT’s generalize the notion

of minimal models introduced in [2]. Their conjecture motivated a great deal of work on

RCFT’s leading in particular to Verlinde algebra [3] and the rich structure they encode

[4]. Moreover it was shown in [5,6] that RCFT’s naturally lead to boundary states, which

in modern terminology we call D-brane states.

In the post duality era, we have learned the importance of D-branes in uncovering

non-perturbative aspects of string theory. Thus it is natural to ask the following question:

We consider strings propagating on a Calabi-Yau background and we vary the moduli of

Calabi-Yau. At points on the moduli which correspond to RCFT’s we naturally have some

finite number of ‘special’ D-branes. What do these D-branes correspond to? How do we

interpret their preferred role at those moduli among the infinitely many allowed D-branes?

Interesting as these question appear, the modern discovery of S-duality raises a further

question: Is the notion of RCFT an S-duality invariant concept? The answer to this is no

[7]. In fact it could hardly be an S-duality invariant concept because the S-dual theory

may not even correspond to string theory so there is no notion of 2d conformal theory on

the S-dual side. Even if the S-dual theory is a string theory, one can easily see that the

RCFT’s on one side do not correspond to RCFT’s on the dual side. For example type

IIB is self-dual, with the roles of fundamental and D-strings exchanged; the rationality

of its toroidal compactifications strongly depends on the BNS but is independent of BR.

However on the S-dual side the roles of BNS and BR are exchanged. Thus rationality is

not an S-duality invariant concept. One might thus consider this concept as not being

a fundamental concept. However, it turns out that at least in some cases the concept of

special points on moduli space makes sense even non-perturbatively. For example the same

considerations that apply to rational points on compactification of strings on T 2 lead to

singling out special points on the moduli space of the type IIB string coupling constant τ

which could potentially have some significance in the full non-perturbative theory. Another

1

way such a concept may remain relevant non-perturbatively is exemplified by compacti-

fication on Calabi-Yau 3-folds. In this case the string coupling constant combines with

other hypermultiplets which come from Kahler moduli (in the type IIB case) and so the

question of rationality, which seems to split between Kahler and Complex moduli, picks

out, in a non-perturbative sense, some special complex moduli on the Calabi-Yau. More

generally, at the very least the moduli corresponding to RCFT’s must be somehow special

at weak string coupling and thus must teach us some extra symmetries about the target

space physics.

In this paper we take up the question of RCFT’s for sigma models on Calabi-Yau

n-folds. More specifically we consider the case of complex dimension 1 in detail and use

it to advance a conjecture about rational points for the more complicated case of sigma

models on Calabi-Yau n-folds.

For the case of (complex dimension one) T 2 target space we uncover the extra sym-

metry principle for the target space for it to correspond to a RCFT. These correspond to

tori which admit Complex Multiplication. This means that there is a complex number λ

(not real) for which z → λz maps T 2 to itself. This is not necessarily an isomorphism,

and is in general a many to one map. Moreover the corresponding Kahler class has to be

a complex multiplier λ for this to correspond to a RCFT with a diagonal modular invari-

ant. Moreover for each Kahler class there exists a canonical complex multiplication which

has the significance that gives the corresponding Cardy states as D0 branes localized at

preimages of a given point on T 2 under this complex multiplication.

In higher dimensions, mathematicians have a generalized notion of complex multi-

plication [8,9,10], which is natural to conjecture is related to the notion of rationality of

conformal theory. This is basically the statement that the mid-dimensional cohomology

and the associated variation of Hodge structure, leads naturally to the period matrix of

a higher dimensional torus and one asks whether the associated torus admits a complex

multiplication. In the case of Calabi-Yau threefolds this leads to a complex torus corre-

sponding to coupling constant matrix of the gauge fields. In the geometric engineering of

N = 2 theories, this is the associates complex torus encoding the BPS masses of N = 2

electric and magnetic charge states. Translated in this way, there is a mathematical conjec-

ture which suggests that in many cases there are only a finite number of points on moduli

where the theory is rational. This is in sharp contrast to the case of T 2 where the rational

points are dense in the space of all conformal theories.

2

The organization of this paper is as follows: In the next section we review a simple

example of c = 1 RCFT based on a circle, which will help us to introduce the notations and

relevant concepts. In section 3 we discuss two families of RCFT based on two-dimensional

tori with extra symmetries: (a) direct product of two circles, and (b) a torus with ZZ3

symmetry. Both examples have been extensively studied in the literature, and we use

them to illustrate general features of rational conformal field theories. Section 4 gives a

friendly introduction into basics of imaginary quadratic number fields, which play a central

role in the characterization of rational CFT’s. Following these introductory sections, in

section 5 we proceed to the general case of c = 2 CFT based on elliptic curve, and formulate

the criteria for CFT to be rational and, further, to be diagonal. Our results allow to classify

such rational conformal field theories. In section 6 we discuss geometric and arithmetic

interpretation of Cardy states in RCFT based on elliptic curve. Finally, in section 7 we

conjecture a generalization of these results to Calabi-Yau manifolds of higher dimension.

In different contexts, relation between string theory and number theory has been

discussed previously in [11,7,12,13,14,15,16].

2. Review of a Compact Free Boson

We start with a review of c = 1 CFT associated with a free bosonic field on a circle of

radius R. Since this theory was extensively studied in the literature, and has been nicely

reviewed in a number of recent papers, see e.g. [17,18,19,20], here we briefly describe only

some aspects that will be relevant in the following sections. In our conventions α′ = 1, so

that the self-dual radius is Rs.d = 1.

For generic values of the radius R, the torus partition function has the following form:

Z(q, q;R) =1

|η|2∑

(p,p)∈Γ1,1

q12 p

2

q12p

2

(2.1)

where q = exp(2πiτ) and Dedekind’s η-function is defined as

η = q1/24∞∏

n=1

(1− qn)

The partition function Z(q, q;R) is given by a sum over the even, self-dual momentum

lattice Γ1,1. Explicitly, the left and the right momenta read:

p =1√2

( nR

+mR), p =

1√2

( nR

−mR)

(2.2)

3

By definition, the theory is rational if one can represent the partition function Z(q, q)

as a finite sum of the form:

Z(q, q) =∑

j,j

Mjjχj(q)χj(q) (2.3)

where Mjj ∈ ZZ≥0 and χi (resp. χj) are holomorphic (resp. anti-holomorphic) characters:

χj(q) = trVjqL0−c/24

For the toroidal examples χi and χj are generalized θ-functions with characteristics.

Modular invariance and existence of the unique vacuum state impose further restric-

tions on the matrix M . Namely, uniqueness of the vacuum implies:

M00 = 1 (2.4)

On the other hand, modular invariance of the torus partition function requires:

[M,T ] = 0, [M,S] = 0 (2.5)

where matrices S and T determine transformation of the characters under SL(2,ZZ):

χi(−1/τ) =∑

j

Sijχj(τ)

χi(τ + 1) =∑

j

Tijχj(τ)(2.6)

The matrix T is diagonal and can be written in terms of the conformal dimensions ∆i and

the central charge c:

Tij = δije2iπ(∆i−c/24) (2.7)

There is no such simple general expression for the matrix S. However, in RCFT’s where

Verlinde algebra is an abelian group algebra Sij are proportional to roots of unity, which

follows from the fact that S diagonalizes Verlinde algebra. For example, in the rational

c = 1 CFT of a compact boson, we have:

Sjj′ =1√2N

e−iπjj′/N (2.8)

For a given RCFT, it is an interesting problem to classify integer matricesM , which satisfy

the relations (2.4) and (2.5), see [21] for a review.

4

Now, let us explain a geometric interpretation of the rationality condition in a theory

of a free compact boson. In other words, we want to analyze when the exponent set I = ibecomes finite. From the explicit form of the left and right momenta (2.2) it is clear that

this happens when R2 is a rational number:

R2 =k

l, k, l ∈ ZZ

where k and l are relatively prime integer numbers. The partition function in this case

reads:

Z(q, q) =∑

i+j=0 mod 2k

i−j=0 mod 2l

χi(q)χj(q)

It is manifestly invariant under T-duality symmetry, which among other things inverts the

radius R and exchanges the winding and momentum modes:

ZZ2 : R↔ 1

R, m↔ n

The chiral primaries in this theory are labeled by index j ∈ ZZ mod 2kl. All theories

with the same value of N = kl have the same fusion ring, which in this simple case is just

the group algebra of:

ZZ2N

Therefore, theories with the same chiral algebra, but different modular invariants corre-

spond to different ways one can decompose N into a product of two integers. In particular,

R2 = integer (or R2 = 1/integer) correspond to diagonal modular invariants Mjj = δjj (or

charge conjugation modular invariants):

Z(q, q) =∑

j∈Iχj(q) χj(q) (2.9)

2.1. Cardy States

Let us now discuss D-branes in rational conformal field theories. Among all consistent

boundary states, there is a special (finite) subset of states, which are invariant under the

full extended chiral algebra. This distinguished set of states, called Cardy states, can be

systematically constructed in a diagonal RCFT following the original work of Cardy [5].

Let us recall that Cardy states are linear combinations of the Ishibashi states obtained

by imposing a modular invariance on the string world-sheet:

|i〉 =∑

i

Sij√S0j

|j〉〉 (2.10)

5

In particular, the number of the Cardy states is equal to the number of the Ishibashi states.

The Ishibashi states, in turn, are defined as generalized solutions to the gluing condi-

tions [6]: (Wn − (−1)∆WΩ(W−n)

)|B〉〉 = 0 (2.11)

where Ω is a gluing automorphism, and ∆W is the conformal dimension of the chiral algebra

operator W . For example, in the case of a free compact boson, the boundary state should

preserve the U(1) current J = i√2N∂X .

Since the Virasoro generators are quadratic in the oscillator modes, the Ishibashi

boundary condition (2.11) is solved by:

(αµn −Rµ

ν αν−n)|B〉〉 = 0 (2.12)

We can also write this condition as:

∂Xµ(z) = Rµν∂X

ν(z) (2.13)

Here we restored space-time indices µ, ν = 1, . . . , d, and R is an automorphism of the

chiral algebra, such that R ∈ O(d) and |det(R)| = 1. In particular, +1 eigenvalues of

the automorphism correspond to Neumann boundary conditions, whereas −1 eigenval-

ues correspond to Dirichlet boundary conditions [17]. For this reason, a boundary state

corresponding to an automorphism with p eigenvalues +1 is referred to as a Dp-brane.

For a given automorphism R, the explicit form of the boundary state |B〉〉 satisfying(2.12) is given by:

|B〉〉 = exp[−

∞∑

n=1

1

nαµ−nRµν α

ν−n

]×∑

|p, p〉 (2.14)

The sum over momenta in this expression goes over all the elements in the momentum

lattice, which satisfy the condition (2.12).

Let us now come back to the case of the diagonal RCFT corresponding to a free boson

on a circle of radius√NRs.d.. In this case, there are only two choices for the automorphism,

R = ±1, corresponding to D1-branes and D0-branes, respectively. In the latter case, we

obtain 2N Ishibashi states, the A-states in the notations of [22,18]:

|An, n〉〉 = exp[+

∞∑

n=1

1

nα−nα−n

]×∑

m

| n√N

+m√N,

n√N

+m√N〉 (2.15)

6

where n ∈ ZZ mod 2N . For the other automorphism one has only two Ishibashi B-states:

|Bn,−n〉〉 = exp[−

∞∑

n=1

1

nα−nα−n

]×∑

| n√N

+m√N,− n√

N−m

√N〉 (2.16)

where n = 0 or n = N .

Using the Cardy’s formula (2.10) and the explicit form for the elements of matrix

S, in this model one finds two boundary states corresponding to D1-branes with different

values (±1) of the Wilson line, and 2N D0-branes located at the equidistant positions on

the circle, see Figure 1:

#(Dp− branes) =

2N p = 02 p = 1

These Cardy states will be our basic building blocks in a specific example of c = 2 RCFT

discussed next.

R 2N D0−branes

Fig. 1: Cardy states in rational c = 1 CFT can be identified with equally spaced

D0-branes (black dots) on a circle.

3. Simple Examples of c = 2 RCFT

Starting with examples in this section, we proceed to our main subject, namely c = 2

RCFT based on the elliptic curve E. Specifically, we analyze the conditions on the complex

structure parameter τ and the complexified Kahler modulus ρ under which the theory

becomes rational and has a diagonal partition function (2.9). These results will help us

build some intuition about what should happen in the general case that will be discussed

in the following sections.

7

3.1. Basic Notations

To begin, we summarize our conventions and recall the discrete symmetries of a general

RCFT based on the elliptic curve (see also [23] for a nice exposition). Throughout the paper

we use the following notations for the real and imaginary parts of τ and ρ:

τ = τ1 + iτ2 =G12

G11+ i

√detG

G11

ρ = ρ1 + iρ2 = B + i√detG

(3.1)

where Gij denote components of the (flat) metric on E. Indeed, it is straightforward to

check thatds2 =

ρ2τ2

|dx+ τdy|2 =

= G11dx2 + 2G12dxdy +G22dy

2(3.2)

Since we are interested in geometric interpretation of RCFT, we further assume that:

τ2 > 0, ρ2 > 0 (3.3)

RCFT based on the elliptic curve E enjoys a large group of discrete symmetries:

PSL(2,ZZ)τ × PSL(2,ZZ)ρ × ZZ2 × ZZ2 × ZZ2 (3.4)

Apart from the last factor, this symmetry group may be viewed as a group of T-dualities.

In particular, the first two factors in this group act via modular transformations on τ and

ρ, respectively. For example:

PSL(2,ZZ)τ : τ 7→ aτ + b

cτ + d

The ZZ2 factors, on the other hand, interchange τ , ρ, and their complex conjugates. Specif-

ically, the first ZZ2 acts as:

ZZ2: (τ, ρ) 7→ (ρ, τ) (3.5)

By analogy with the corresponding symmetry of higher dimensional varieties, we refer to

this ZZ2 as to the mirror transform.

The second ZZ2 factor in (3.4) is space-time parity transformation acting as follows:

ZZ2: (τ, ρ) 7→ (−τ ,−ρ) (3.6)

8

Finally, the last ZZ2 factor in (3.4) reverses world-sheet orientation:

ZZ2: (τ, ρ) 7→ (τ,−ρ) (3.7)

3.2. A Product of Two Circles: E = S1 × S1

Our first simple example of c = 2 RCFT will be a product of two c = 1 rational CFT’s

corresponding to a product of two circles of radii:

R1 =

√k1l1

and R2 =

√k2l2

where ki and li are some (pairwise co-prime) integers. In these models both τ and ρ

are pure imaginary (modulo real integer part, which can be set to zero by PSL(2,ZZ)

transformations):

τ = i

√k1l2k2l1

, ρ = i

√k1k2l1l2

Note, that both τ and ρ satisfy quadratic equations with integer coefficients (and the same

discriminant).

Using the analogous result for a single compact boson, we find that RCFT based

S1 × S1 has a diagonal modular invariant if l1 = 1 and l2 = 1, i.e. up to modular

transformations:

τ = i

√k1k2, ρ = i

√k1k2

Another way to describe these τ and ρ is to say that τ is a solution to the quadratic

equation:

k2τ2 + k1 = 0 (3.8)

while ρ is an integer multiple of τ :

ρ = k2τ

This (strange) relation between τ and ρ is a precursor of the general property of the elliptic

curve, called complex multiplication. As we will see in the following sections, all elliptic

curves with this property correspond to a diagonal RCFT, and the converse is also true.

Since in the present example, the RCFT is a product of two theories, the Verlinde

algebra is just a product:

ZZ2k1× ZZ2k2

Note, that the total dimension of the chiral ring is equal to 4k1k2. This also gives the

number of D0-branes in this theory. Indeed, boundary states corresponding to D-branes in

9

the c = 2 RCFT in consideration can be obtained by tensoring the suitable Cardy states

in the two copies of c = 1 RCFT. Specifically, a product of Dp1 boundary state with Dp2

boundary state gives a D(p1 + p2)-brane boundary state:

|D(p1 + p2)〉 = |Dp1〉 ⊗ |Dp2〉

If we tensor two A-type Cardy states, we obtain boundary states corresponding to

D0-branes, 4k1k2 in number. The D0-branes are distributed on a torus in a regular lattice

of 2k1 rows and 2k2 columns, as shown in Figure 2. On the other hand, if we tensor two

B-type boundary states, we get four D2-branes, which cover the entire torus and differ in

the values of Wilson lines they carry. Specifically, since each D1-brane on a circle carries

±1 Wilson line, by tensoring two of them we get four boundary states, labeled by (±1,±1).

Finally, tensoring A-type Cardy states with B-type Cardy states gives D1-branes

parallel to the sides of the torus. Namely, if we tensor D0-brane state on the first circle

with a D1-brane state on the second circle, we obtain D1-branes wrapped on the second

circle inside E = S1 × S1. Since we could take 2k1 boundary states corresponding to

D0-branes and there are 2 possible choices for the Wilson line on the D1-brane, in total we

get 2× 2k1 = 4k1 parallel D1-branes. Similar arguments show that there are 4k2 parallel

D1-branes wrapped on the other basic cycle of the torus, see Figure 2. Hence, there are

4(k1 + k2) D1-branes in total.

Summarizing, we have:

#(Dp− branes) =

4k1k2 p = 04(k1 + k2) p = 14 p = 2

(3.9)

D0−branes D1−branes1

τ

Fig. 2: Cardy states in rational CFT based on a torus E = S1 × S1.

10

3.3. SU(3) Torus

There is another simple example of c = 2 RCFT based on the torus that has been

extensively studied in the literature [22,24,25]. It is SU(3) WZW theory at level 1 corre-

sponding to elliptic curve E with extra ZZ3 symmetry [26]:

τ = ρ = exp(2πi/3)

As in the previous example, in this case both τ and ρ also satisfy a quadratic equation:

τ2 + τ + 1 = 0 (3.10)

The dimension of the chiral ring of this theory is equal to 3, and the Verlinde algebra

is just a group algebra of:

ZZ3

Comparing this with the previous example, one might expect that in general the fusion

rules are given by (a product of) cyclic groups. In what follows we will show that this is

indeed always the case; specifically, the Verlinde algebra is a group algebra of:

ZZn1× ZZn2

(3.11)

This guess includes the special case of a single cyclic group (as in the present example)

when one of the factors is trivial, ni = 1.

Cardy states corresponding to various D-branes in this model have been studied in a

number of papers [22,24,25,27]. Here, we summarize the result:

#(D0− branes) = 3

#(D2− branes) = 1(3.12)

The number of D0-branes is expected to be 3 on general grounds. In fact, in all theories

the number of Cardy states corresponding to D0-branes should be equal to the dimension

of the chiral ring, which is indeed 3 in the present case.

Since generic elliptic curve E does not reduce to a product of two circles, at the

moment we have nothing to say about D1-branes. We can just briefly mention that certain

boundary states in this model, studied in [22,24,25,27], can be identified with D1-branes

wrapped on the shortest cycles of the torus rotated by π/3, as illustrated on Figure 3. In

the general discussion below we will find the complete set of boundary states in this theory.

A little bit more interesting is a result for D2-branes. Combining it with the result of

the previous example, one might conclude that the number of D2-branes can be at least

1 or 4. Quite surprisingly, we will show that this a general answer: in all c = 2 RCFT’s

based on elliptic curve, there is either one or four D2-branes. The number depends on

certain arithmetic properties of E.

11

D0−branes D1−branes

τ

12π/3

Fig. 3: D-branes on the SU(3) torus.

4. Imaginary Quadratic Number Fields

Before we discuss the general case of (rational) conformal field theory based on the

elliptic curve E we need to introduce a few basic notions from number theory that will

naturally appear in our discussion. In fact, some of these objects have already entered our

discussion in the implicit form. For example, in two special cases discussed in the previous

section we have noticed that complex parameter τ satisfies a quadratic equation of the

form, cf. (3.8) and (3.10):

aτ2 + bτ + c = 0 (4.1)

with relatively prime integer coefficients a, b, and c. We shall call this quadratic equation

a minimal polynomial for τ , and denote by D its discriminant:

D = b2 − 4ac (4.2)

In all cases relevant to physics D < 0, so that τ has a non-zero imaginary part:

τ =−b+

√D

2a(4.3)

Mathematically, it means that τ is valued in the imaginary quadratic number field:

K =Q(√D)

This particular way of writing the number field K indicates that it can be obtained from

the familiar field of all rational numbers, Q, by introducing√D. In other words, every

number x in Q(√D) can be written in the form:

x = α+ β√D

12

where α and β are rational numbers, α, β ∈Q. Notice, the way we construct the number

field Q(√D) from rational numbers is very similar to how one usually defines the field of

complex numbers, C, supplementing the field of real numbers, IR, with√−1.

In our applications, τ is not just a number – it is a modulus of the elliptic curve:

E =C/(ZZ⊕ τZZ) (4.4)

It turns out that elliptic curves with modular parameter τ ∈Q(√D) have a nice property

called complex multiplication [28,29,30]. Elliptic curves with this property enjoy a lot of

wonderful arithmetic and geometric properties.

To explain what complex multiplication means, let us consider endomorphisms of the

elliptic curve E, i.e. holomorphic maps from E to itself:

ϕ : E → E

Note, ϕ is a finite degree map (not necessarily degree one). To describe such maps more

explicitly, we can view the elliptic curve E as a quotient of a complex plane (parameterized

by z) by a lattice ZZ ⊕ τZZ, cf. (4.4). Then, an endomorphism ϕ simply acts as z 7→ ϕz.

Since ZZ ⊕ τZZ is a two-dimensional lattice, we have only have to verify that ϕ maps its

generators to some other other elements in this lattice:

ϕ · 1 = m1 + n1τ,

ϕ · τ = m2 + n2τ(4.5)

Clearly, any elliptic curve has many trivial endomorphisms corresponding to multiplication

by an integer, ϕ ∈ ZZ. In order to see if there exist any non-trivial endomorphisms, one

can take ϕ from the first equation and substitute it to the second equation. As a result,

one finds a quadratic equation with integer coefficients of the form (4.1). This simple

calculation illustrates that elliptic curves with non-trivial endomorphisms have τ in some

imaginary quadratic field. In fact, it turns out that an elliptic curve E has a non-trivial

endomorphism if and only if τ obeys a quadratic equation (4.1) with integer coefficients.

In this case, E is said to have complex multiplication (or to be of CM-type). Summarizing,

the endomorphism ring of a general elliptic curve can be one of the following:

End(E) =

ZZ, no CM

ZZ+ ZZaτ, CM-type, τ = −b+√D

2a

(4.6)

13

Thus, complex multiplication gives another way to characterize elliptic curves with such

‘special’ values of τ .

There is a close relation between imaginary quadratic τ , which obey (4.1), and binary

quadratic forms: (2a bb 2c

)

In our discussion, such forms will be associated with intersection form of lattices. Notice,

the form above naturally defines a two-dimensional even lattice. However, this form is not

unique. Namely, for any S ∈ SL(2,ZZ), the intersection form:(2a′ b′

b′ 2c′

)= S

(2a bb 2c

)Str

defines the same lattice. The invariant associated with such a lattice is the discriminant

D = b2 − 4ac. For a given value of D, the equivalence classes of the integral binary

quadratic forms form a finite abelian group, the so-called class group [31,28,32]:

Cl(D)

The order of this group |Cl(D)| = h(D) is called the class number. It is naturally identified

with the number of ideal classes, h(K), of the imaginary quadratic field K =Q(√D).

The last object we need to introduce is an order, Of .

Of = ZZ⊕ ZZ[faτ ] (4.7)

Here, a a leading coefficient in the quadratic polynomial (4.1) for τ , and f is a positive

integer number, called a conductor of Of . We can view an order Of as a two-dimensional

lattice in the number fieldK =Q(√D), generated by 1 and faτ . The reasonOf will appear

in our discussion is that any element ϕ ∈ Of obviously gives a complex multiplication, cf.

(4.5). Note also, that the endomorphism ring itself, End (K) is also an order with1 f = 1.

5. General c = 2 RCFT Based on the Elliptic Curve E

In this section we are going to study general c = 2 conformal field theory based on

elliptic curve with arbitrary parameters τ and ρ. We will show that CFT is rational if (and

only if) both τ and ρ take values in the same imaginary quadratic field. The discriminant

of this field gives the dimension of the chiral ring. We also show that a condition for

diagonal modular invariant implies a further relation between τ and ρ. Namely, one has

to be a complex multiplication for the other.

1 This special order in K is called the ring of integers.

14

5.1. Momentum Lattices

In general, the partition function of c = 2 CFT based on elliptic curve E =C/(ZZ⊕τZZ):

Z(q, q) =1

η2η2

∑

(p,p)∈Γ2,2

q12p2

q12p2

(5.1)

is given by the sum over even, self-dual momentum lattice:(pp

)∈ Γ2,2 =

i√2τ2ρ2

ZZ

(11

)⊕ ZZ

(ρρ

)⊕ ZZ

(ττ

)⊕ ZZ

(ρτρτ

)(5.2)

This lattice will be one of the central objects in our discussion, and, as we shall see in

a moment, properties of the CFT, like rationality etc., can be formulated and analyzed

in terms of momentum lattices. For this reason, it is convenient to introduce a few more

objects associated with the momentum lattice Γ2,2.

First, we can define the following sublattices in Γ2,2. Let Γ0 be the lattice of left-

moving momenta p for a fixed value of p (say, for p = 0) and a similar lattice Γ0:

Γ0 = p |(p0

)∈ Γ2,2

Γ0 = p |(0p

)∈ Γ2,2

(5.3)

Since Γ2,2 is even integer lattice, the same is true about Γ0 and Γ0. The rank of these

lattices is not greater than 2, and generically it is zero.

By simply forgetting the right (or left) momentum, we can also define the following

projections:

ΓL = p |(p∗

)∈ Γ2,2

ΓR = p |(∗p

)∈ Γ2,2

(5.4)

Both ΓL and ΓR can be characterized as sets where p (or p) take their values. In general,

unlike (5.3), ΓL,R are not lattices. Of course, in some special cases it may happen that

the rank of ΓL,R is less than 4. As we shall see below, these are precisely the occasions

relevant to rational theories.

Note, from the above definitions we obtain straightforward relations:

Γ0 ⊆ ΓL

Γ0 ⊆ ΓR

Γ0 ⊕ Γ0 ⊆ Γ2,2

(5.5)

15

5.2. A Criterion for Rationality

There are various ways of defining rational CFT’s. In our examples associated with

tori, it is convenient to formulate the condition for rationality in terms of momentum

lattices: a CFT is rational if and only if the left momentum lattice Γ0 is a finite index

sublattice in ΓL. In such cases, both Γ0 and ΓL are rank two sublattices in Γ2,2. Moreover,

it is easy to see that they are dual lattices:

Γ0∼= Γ∗

L, Γ0∼= Γ∗

R (5.6)

Indeed, since Γ2,2 is even, self-dual integer lattice, for any vector (q, q) ∈ Γ2,2 and a given

vector (p, 0) ∈ Γ0 we have a pairing:

(p, 0) · (q, q) = pq ∈ ZZ

Therefore, any vector in Γ0 also belongs to Γ∗L. Conversely, to show Γ∗

L ⊆ Γ0 let us take

a vector p ∈ Γ∗L. Then, using the above equation and self-duality of Γ2,2, one finds that

(p, 0) is a vector in Γ2,2. Hence, Γ∗L∼= Γ0. Similar arguments give the second isomorphism

in (5.6).

Therefore, we conclude that the study of rational conformal field theories based on

the elliptic curve E is related to the study of integer even two-dimensional lattices. In

particular, it gives a classification of such RCFT’s. We will come back to this later.

Now, let us discuss the geometric properties of the elliptic curve corresponding to

rational conformal field theory. Suppose that a CFT associated with E is rational, i.e. Γ0

is a finite index sublattice in ΓL. Using the explicit expression (5.2) for the right momenta

p, we find that the elements of Γ0 correspond to integer numbers (m1, m2, n1, n2) ∈ ZZ4,

which obey two independent linear relations [7]:

m1 +m2ρ+ n1τ + n2τρ = 0

m′1 +m′

2ρ+ n′1τ + n′

2τρ = 0(5.7)

If we solve, for example, for ρ from the second equation and substitute the result to the

first equation, we find a quadratic equation for τ , with integer coefficients2:

aτ2 + bτ + c = 0, gcd(a, b, c) = 1 (5.8)

2 Alternatively, we could use the fact that any three momentum vectors p ∈ ΓL (or p ∈ ΓR)

satisfy a linear relation over Q, since Γ0 is a ZZ⊕ ZZ-module.

16

with discriminant D = b2 − 4ac. Since imaginary part of τ has to be strictly positive, cf.

(3.3), we conclude that τ has to belong to imaginary quadratic number field K =Q(√D):

τ =−b+

√D

2a(5.9)

If we now substitute this τ back into (5.7), we find that ρ is linear in√D over Q.

Hence, both τ and ρ are elements in K. In order to show that the converse is also true, one

can take τ, ρ ∈ K, and construct, for example, τ and τρ in terms of 1 and ρ. Since both τ

and ρ are assumed to be linear functions of√D with rational coefficients (with τ2 > 0 and

ρ2 > 0), one can always write τ and τρ as linear functions of ρ with rational coefficients.

Multiplying the resulting relations by a suitable integer, one finds two equations of the

form (5.7) with integer coefficients, where n2 = 0 and n′1 = 0. Therefore, by construction

these relations are independent and define a lattice Γ0.

Summarizing, we obtain an effective criterion for rationality of c = 2 conformal field

theory based on the elliptic curve E:

RCFT ⇐⇒ τ, ρ ∈Q(√D)

In other words, we found that in order for CFT to be rational, both the target space

torus and its dual should have complex multiplication relative to the same quadratic imag-

inary field.

5.3. Dimension Of The Chiral Ring And Verlinde Algebra

As we found in the previous subsection, rational conformal field theories based on the

elliptic curve E are naturally attached to even integer lattices. Specifically, let vi, i = 1, 2,

be the generators of the momentum lattice Γ0. Since the intersection form is even, we can

write it as:

vi · vj = f

(2a bb 2c

)(5.10)

for some integer numbers a, b, c, and f , such that gcd(a, b, c) = 1.

By definition, the dual lattice ΓL = Γ∗0 is generated by vectors v∗i with intersection

form:

v∗i · v∗j =1

fD

(−2a bb −2c

)(5.11)

where D = b2−4ac is the discriminant. It is clear from (5.10) and (5.11) that the dimension

of the chiral ring, given by the index [ΓL : Γ0], is equal to:

[ΓL : Γ0] = f2|D|

17

Notice, that the right hand side is expressed in terms of invariant quantities.

Furthermore, the Verlinde algebra is the group algebra of:

D(Γ0) = Γ∗0/Γ0 (5.12)

In mathematics literature this group is usually called the discriminant group [33,34], see

also [32]. It is a finite abelian group of order f2|D|. Since Γ0 is a lattice of rank two, in

general, the discriminant group is a product of two cyclic groups, in agreement with what

we found in the specific examples of tori with extra symmetries, cf. (3.11). Specifically,

D(Γ0) is generated by two elements, g and h, such that:

g2afhbf = 1 and gbfh2cf = 1

The structure of this group depends in a crucial way on the arithmetic. Specifically,

D(Γ0) =

ZZf × ZZfD, D = 1 mod 4ZZ2f × ZZ2fD′ , D = 0 mod 4 (D′ = D/4), b 6= 0ZZ2fa × ZZ2fc, b = 0

This gives a general characterization of the Verlinde algebra in the rational conformal

field theory based on the elliptic curve E.

5.4. Diagonal Modular Invariants

Now we turn to the main problem, namely, analysis of the conditions under which

the partition function (5.1) takes the diagonal form (2.9). As we explained in the previous

subsections, curve E associated with a rational CFT has complex multiplication. However,

the ring End(E) itself did not enter our discussion so far. Here, we show that RCFT has

a diagonal modular invariant iff either ρ or τ (or modular transformations thereof) belong

to End(E), i.e. iff ρ is a complex multiplication for a given τ , up to discrete symmetries

(3.4).

Diagonal modular invariant essentially implies identification of left and right momen-

tum lattices. Namely, given an even integer lattice Γ0, one can canonically reconstruct the

whole momentum lattice Γ2,2, which is even and self-dual. Specifically, we take two copies

of ΓL = Γ∗0:

Γ2,2 = (ΓL,Γ′L) (5.13)

with the equivalence relation:

ΓL − Γ′L = Γ0 (5.14)

18

to see that the lattice (5.13) constructed in this way is even, let us take a vector (p, p) ∈ Γ2,2.

By the equivalence relation (5.14) we have p = p+v, where v ∈ Γ0, and p ∈ Γ∗0. Therefore,

(p, p) · (p, p) = p2 − p2 = (p+ p)(p− p) =

= −v(2p+ v) = −2vp− v2(5.15)

Since Γ0 is taken even, both terms here are even and the claim follows. Furthermore,

to show that Γ2,2 constructed in (5.13) is self-dual, one can provide a basis, (pi, pi) =

(v∗i , v∗i ), (vj, 0), where vj are the generators of Γ0, and v∗i ∈ Γ∗

0 are the dual generators.

Then, the bilinear form looks like

(pi, pi) · (pj , pj) =(0 1

1 ∗

)

The determinant of this matrix is clearly equal to 1. Therefore, Γ2,2 is self-dual.

Now we want to compare a general lattice of the form (5.11) with the momentum

lattice (5.2). Up to discrete symmetries (3.4) we can choose τ and ρ, such that the left

momentum lattice is generated by (vectors proportional to) 1 and τ :

Γ∗0 = ΓL = i√

2τ2ρ2,

iτ√2τ2ρ2

It is easy to compute the intersection form of this lattice:

v∗i · v∗j =

(1

2τ2ρ2

τ12τ2ρ2

τ12τ2ρ2

|τ |22τ2ρ2

)

Since Γ0 has to be an even integer lattice, one should be able to write this intersection as

(5.11). Comparing the individual entries, we find that in diagonal RCFT τ and ρ look like

(up to discrete symmetries (3.4)):

τ =−b+

√D

2a, ρ = faτ (5.16)

Hence, ρ is a complex multiplication for an imaginary quadratic τ . More precisely, ρ should

be associated with a generator of an order in the imaginary quadratic field K =Q(√D):

ρ ∈ Of

with the conductor f .

19

It is easy to verify that the converse is also true. Namely, given τ and ρ of the form

(5.16), the corresponding modular invariant is diagonal. Indeed, substituting (5.16) in the

formulas for the momentum vectors (5.2), we find that the momentum lattice Γ2,2 is of

the form (5.13). Specifically, we have:

−i√

2afτ2 p = m1 +m2ρ+ n1τ + n2ρτ =

= (m1 + fcn2) + (n1 − afm2 + bfn2)τ(5.17)

It immediately follows that:

ΓL = (p, ∗)T ∈ Γ2,2 =i√

2afτ2(ZZ⊕ ZZτ) (5.18)

In order to find elements of Γ0, we have to solve p = 0. For the right momenta we

find:−i√

2afτ2 p = m1 +m2ρ+ n1τ + n2ρτ =

= (m1 − fcn2) + (n1 + afm2 − bfn2)τ(5.19)

Hence, p = 0 gives two conditions:

m1 = fcn2

n1 = −afm2 + bfn2

Substituting this into (5.17) yields:

p|p=0 =if√2afτ2

((2aτ + b)(−m2) + (2c+ bτ)n2

)∈ Γ0

Therefore, we explicitly constructed Γ0:

Γ0 = (p, 0)T ∈ Γ2,2 =if√2afτ2

(ZZ[2aτ + b]⊕ ZZ[2c+ bτ ]) (5.20)

The resulting lattices ΓL, Γ0, and Γ2,2 are related as in (5.13) – (5.14), so that the cor-

responding CFT is diagonal. It is straightforward to check this directly computing the

partition function (5.1) for these momentum lattices. As a result, one finds a diagonal

modular invariant:

Z(q, q) =∑

ω∈Iχω(q)χω(q)

where the exponent ω ∈ I that labels representations of the chiral algebra can be identified

with left momenta, cf. (5.12):

ω ∈ ΓL/Γ0 (5.21)

20

and the characters χω(q) have the form:

χω(q) =1

η2

∑

v∈Γ0

q12 (v+ω)2 (5.22)

Finally, we can combine all the results in this section to conclude that diagonal c = 2

rational conformal field theories are classified by the following data:

i) discriminant D = 0, 1 mod 4 (a negative integer, such that (−D) is square-free);

ii) conductor f (a positive integer);

iii) an element of the class group Cl(D).

In terms of this data, the CFT has chiral ring of dimension |D|f2.

Suppose, for example, we want to know how many diagonal RCFT’s have the chiral

ring of dimension 163. Using the above results, we can immediately answer this question.

Since h(163) = 1, the answer is surprisingly simple: there is a unique RCFT with this

property. On the other hand, if we asked a similar question about RCFT’s with chiral

ring of dimension 159, we would find h(159) = 10 such theories. This simple example

illustrates sporadic pattern of RCFT’s, which nevertheless can be completely explained by

the methods of number theory.

Note, it also follows from our analysis that the set of c = 2 rational conformal field

theories on tori is dense.

5.5. A Digression: The First Main Theorem of Complex Multiplication

In this subsection (which can be skipped, especially in a single reading) we digress on

another remarkable property of elliptic curve with complex multiplication. Throughout

the paper we mainly viewed the elliptic curve as a quotient of C by a two-dimensional

lattice. However, we could also view E as an algebraic curve defines, say, by a Weierstrass

polynomial:

E: y2 = 4x3 − g2x− g3 (5.23)

where g2 and g3 are related to the modular parameter τ via invariant j-function:

j(τ) =1728g32g32 − 27g23

=θE8

(τ)3

η(τ)24=

= q−1 + 744 + 196844q + 21493760q2 + 864299970q3 + . . .

(5.24)

Notice, that the coefficients in power series expansion of the j-function are integer numbers.

21

Even though j(τ) is a very non-trivial function of τ , there is a nice characterization

of its values for τ ∈ Q(√D), known as the first main theorem of complex multiplication.

Namely, suppose elliptic curve E has a complex multiplication, i.e. τ obeys a quadratic

equation (4.1). Then, j(τ) also obeys a polynomial equation with integer coefficients of

degree h (where h = h(D) is the class number of the field Q(√D)):

P (z) = zh + a1zh−1 + . . .+ ah = 0, ai ∈ ZZ (5.25)

A solution to such equation is called an algebraic integer 3. Therefore, j-invariant of elliptic

curve E with complex multiplication is an algebraic integer and E is naturally defined over

the number field K(j(τ)).

Motivated by this nice result, one might expect that a proper criterion for CFT to be

rational should be formulated as a condition on the algebraic variety to be defined over

the algebraic closure Q, obtained from the field Q by adjoining the roots of all irreducible

polynomials like (5.25). It is easy to see, however, that this criterion would be wrong.

Indeed, it would predict “too many” RCFT’s. For example, in the case of Calabi-Yau

manifolds it would predict existence of infinitely many points (which are dense) in the

moduli space, whereas in the later sections we will argue to the contrary.

More elementary is to see that the above criterion fails already in the case of the

elliptic curve E. Indeed, in general, the converse of the first main theorem of complex

multiplication is not true, so it can not be formulated as “if and only if” condition. On

the other hand, from the analysis of the previous sections, we know that CFT is rational

if and only if E has complex multiplication. This demonstrates that the right signature of

the rational CFT is complex multiplication, rather than a possibility to define the target

space variety over Q.

6. Geometric Interpretation of Cardy States

In the previous sections we have established a relation between RCFT data and arith-

metic of the elliptic curve E. Motivated by such a relation, one may wonder if it can be

extended to string theory, including D-branes and other non-perturbative objects. Due to

3 The first main theorem of complex multiplication further says that if z = j(τ) is one of these

numbers, then K(j(τ)) is the maximal abelian extension of K =Q(√D), with Gal(K(j(τ))/K) =

Cl(End(E)) acting transitively on the set of numbers j(τ) [28,31].

22

their geometric nature, D-branes seem to be especially promising. In the weak coupling

limit they can be viewed as submanifolds of E of various (co)dimension.

From the CFT point of view, there are some ‘special’ D0-branes, which preserve the

full chiral algebra. The corresponding boundary states were explicitly constructed by

Cardy [5]. Therefore, one could ask: “What is arithmetic/geometric interpretation of the

Cardy states?” There are several natural candidates for the answer to this question. For

example, once we deal with arithmetic of E, one might consider rational points of E, i.e.

solutions of (5.23) with rational values of the coordinates x, y ∈Q. However, these can not

correspond to Cardy states. Indeed, the number of rational points on E may be infinite,

whereas the number of Cardy states is always finite (and equal to the dimension of the

chiral algebra):

#(Cardy states) = |D|f2

where D is the discriminant of the quadratic polynomial (5.8) for τ , and f is the conductor.

In this section we study D-branes on elliptic curve E with diagonal modular invariant:

Z(q, q) =∑

j∈Iχj(q) χj(q) (6.1)

where the exponent j can be identified with momentum, cf. (5.21):

j ∈ ΓL/Γ0 (6.2)

and the characters have the form (5.22). Following Cardy [5], we show that there are

always |D|f2 D0-branes in this theory, which correspond to the regular points of ΓL/Γ0.

On the other hand, the number of D2- and D1-branes depends on the arithmetic of the

elliptic curve in a very interesting way. For example,

#(D2− branes) =

1 Df odd4 Df even

Note, simple examples studied in section 3 agree with this general result, see (3.9) and

(3.12). Below we explain these results in more detail.

Before we start, let us remind that Cardy states are linear combinations of Ishibashi

states (2.14):

|B〉〉 = exp[−

∞∑

n=1

1

nαµ−nRµν α

ν−n

]×∑

|p, p〉 (6.3)

23

which satisfy boundary conditions (2.12). It is convenient to write this boundary condition

in the following form:

p = −Rp (6.4)

Since labels j in (6.2) are identified with momenta, we can say that for a given R the

Ishibashi states are labeled by:

(j,−Rj), j ∈ ΓL/Γ0

However, since we assume that RCFT is diagonal (6.1), the Ishibashi state (j, j) appears

in the closed string spectrum only if j = j:

(j, j) = (j,−Rj)

It follows that for a given gluing condition (6.4) the Ishibashi states are labeled by fixed

points of R in the exponent set I:

j = −Rj, j ∈ ΓL/Γ0 (6.5)

In particular, the number of solutions to this conditions give us the number of the corre-

sponding Dp-branes. In the following subsections we solve (6.5) for each value of p. Note,

that j = 0 is always a solution to (6.5) for any allowed R, i.e. there is always at least one

corresponding D-brane.

6.1. D0-branes

In order to get a D0-brane, we have to impose Dirichlet boundary conditions in both

spatial directions, so that 2× 2 matrix R must have two eigenvalues −1, i.e.:

R = −1

In this case, the boundary condition (6.4) has the

p = p

It does not impose any further constraints on the momentum, so that and p ∈ ΓL/Γ0 is a

solution. Hence, the number of D0-branes is given by the following universal formula for

all models:

#(D0− branes) = |D|f2 (6.6)

24

This number is the same as the dimension of the chiral ring, and suggests interpretation of

D0-branes as special points on the torus E. Indeed, we can think of D0-branes as regular

points in the quotient ΓL/Γ0 or else, as preimages of a marked point on a torus E under

a specific complex multiplication 4 (see Figure 4):

z 7→ ρ · z

where, cf. (5.16):

ρ = f(2aτ + b) (6.7)

Indeed, one can easily check that ρ · 1 = f(2aτ + b) and ρ · τ = −f(2c + bτ) are the two

generators of the lattice Γ0, cf. (5.20).

1

2c+bτ

2a

τ

τ+b

Fig. 4: D0-branes in rational CFT can be identified with points (black dots) in

the lattice 〈1, τ〉 modulo 〈ρ, ρτ〉. In this figure we illustrate this for a specific model

where τ satisfies the quadratic equation: τ2 + 2τ + 2 = 0. In this example, a = 1,

b = 2, c = 2, and D = −4.

4 The complex multiplication ρ has a number of special properties. First, note that it can be

obtained by taking a derivative of the defining quadratic polynomial for τ :

ρ = fdQ(τ)

dτ, Q(τ) = aτ2 + bτ + c

Another distinguished property of ρ is that it corresponds to a complex multiplication whose

square is multiplication by an integer. Moreover, it is the only complex element (up to integer

multiples) in the End(E) with this property. Explicitly, the square of the element 2aτ + b is:

(2aτ + b)2 = 4a2τ2 + 4abτ + b2 = 4a(aτ2 + bτ) + b2 = 4a(−c) + b2

which is the discriminant, D.

25

Geometrically, it is convenient to visualize the set ΓL/Γ0 as a parallelogram with edges

f(2aτ + b) and f(2c+ bτ) in the lattice ΓL.

Note that ρ = f(2aτ + b) is closely related to the Kahler structure of the torus, which

is ρ = faτ . In fact, if bf is even ρ can be viewed as twice the Kahler class (with a

suitable shift of ρ by bf/2. Thus roughly speaking the Kahler class defines the relevant

endomorphism of the torus which defines the Cardy states by its preimage.

6.2. D2-branes

The next simplest case is when we impose Neumann boundary conditions in all the

directions:

R = 1

The involution R inverts the exponent set I = ΓL/Γ0:

R : p 7→ −p (6.8)

and flips the parallelogram subtended by vectors f(2aτ + b) and f(2c+ bτ), as shown in

Figure 5.

R

f(2c+b+b)τf(2a

τ)

Fig. 5: In the case of D2-brane boundary conditions, the involution R flips the

parallelogram made by vectors f(2aτ + b) and f(2c+ bτ). The origin (black dot)

is always fixed under this involution. The other three potential fixed points are

denoted by empty circles.

According to (6.5), the Ishibashi states are in one-to-one correspondence with the

fixed points of this involution, modulo the lattice Γ0. Geometrically, it is clear that there

is either one or four fixed points on the parallelogram (see Figure 5), depending on whether

its edges have odd or even coordinates in the lattice ΓL∼= 1, τ. Indeed, the origin is

26

always a fixed point of R. Let us see when there are extra fixed points. The potential

candidates are middle points on the edges of the parallelogram and a point in the middle

(denoted by empty circles in Figure 5). The explicit coordinates of these points in the

lattice ΓL∼= 1, τ are the following:

(fb/2, fa), (fc, fb/2), (fc+ fb/2, fa+ fb/2)

It is clear that all of these points are in the lattice Γ0 if and only if both fb is even. Hence,

we arrive to the following general result:

#(D2− branes) =

1 bf odd4 bf even

(6.9)

Note that in the case of bf even, the four inequivalent D2 branes differ by the value of Z2

Wilson lines along the two cycles.

6.3. D1-branes

Finally we consider the case of D1-branes. In this case we have one Neumann and

one Dirichlet direction, which correspond to +1 and −1 eigenvalues of R, respectively.

Allowing for D1-branes of arbitrary orientation, we can write the corresponding involution

R as:

R : p 7→ αp∗ (6.10)

where α is some phase, |α| = 1. Note that this is an order 2 operation. Since p takes

values in the lattice ΓL∼= 1, τ, the involution R must respect this lattice. In particular

it should map a basis of ΓL into another basis:

1 7→ − A−Bτ

τ 7→C +Dτ(6.11)

where A, B, C, D are integer numbers, such that AD − BC = 1. Therefore, we get two

conditions:α =− A−Bτ

ατ =C +Dτ(6.12)

from which we can eliminate α:

−τ =A+Bτ

C +Dτ(6.13)

Simply put, the last condition says that −τ should be an involution of PSL(2;ZZ) acting

on τ , for otherwise we wouldn’t have any D1-branes. Then, for every τ , which satisfy a

relation of the form (6.13), there might be different involutions corresponding to different

α’s. Therefore, it is natural to split the question in two parts, and classify first all τ which

solve (6.13), and then classify all possible α’s.

27

|τ|=1(τ)=0Re

4f(a+c)

12f

4fa

12f

Fig. 6: Two families of the solution for τ lie either on a unit circle or on the

imaginary line. There are also two special cases, τ = i and τ = exp(2πi/3),

represented by black dots. A number near every point indicates the total number

of D1-brane boundary states in the corresponding RCFT.

It is easy to see that the only τ (in the fundamental domain), which solve (6.13), are

(i) either pure imaginary, τ1 = 0, or (ii) lie on a unit circle, |τ | = 1, see Figure 6. To

see that these are the only solutions, with no loss of generality we can assume τ is in the

fundamental domain of the upper half plane. We are looking for involutions of SL(2, Z)

acting on it which give −τ . On the other hand −τ is also in the fundamental domain

of the upper half plane. This can only be consistent with the notion of the fundamental

domain if the involution is ± the identity matrix, or it is ±S in which case it maps the

boundary of the fundamental domain to itself. There is one extra case corresponding to

the involution TST−1 which fixes τ = exp(2πi/3). In the first case we have τ = −τ which

states that τ is pure imaginary. In the latter case it implies that τ is on the unit circle.

The two different families of the solutions for τ correspond to elliptic curves with

different symmetries, simple examples of which were discussed in section 3. Let us now

consider each of the two cases in turn:

(i) – a product of two circles

In the first case b = 0 and the solutions to (6.13) look like:

τ =

√− c

a

28

In fact, this is precisely the case discussed in section 3.1, where the torus is a product of

two circles:

E = S1 × S1

All possible D1-branes in this case were classified in (3.9), with k1 = fa and k2 = fc:

#(D1− branes) = 4f(a+ c) (6.14)

and come from α = 1 or α = −1. They correspond to 2fa equally spaced D1 branes along

one direction and 2fc equally spaced D1 branes along the other direction of the torus.

Moreover each of these one branes can have a ZZ2 Wilson line on them.

(ii) – a torus symmetric relative to the diagonal

In this case |τ | = 1 (which implies a = c) and the torus E has extra symmetry

corresponding to the reflections relative to the diagonals. In particular z → αz is a

symmetry when α = ±τ , which thus generically yields two involutions R. For the case of

τ = i we have 4 involutions given by z → ikz. For τ = e2πi/3 we have 6 involutions given

by z → ωkz where ω is a 6-th root of unity. The corresponding fixed characters under this

involution will correspond to equally spaced D1 branes in the direction given by√α.

The corresponding values of τ are:

τ =−b+

√D

2a, D = b2 − 4a2

For the generic case, the total number of D1-branes, which is just the number of the

fixed lattice points under the two reflections is

#(D1− branes) = 4fa (6.15)

For the case τ = i the D1 branes making angles multiple of π/4. For D1 branes in the

direction 2nπ/4 (n = 0, 1) we have 2f equally spaced branes each of which can have an

extra Z2 Wilson line. For D1 branes in the directions π/4, 3π/4 we have 2f equally spaced

branes (all branes passing through the origin) without any extra possibility of Wilson

lines. The D1 branes corresponding to the latter angles do not come from tensoring

Ishibashi states of the two decoupled circles, but rather it corresponds to using the extra

Z2 exchanging the two circles. For τ = e2πi/3 we have D1 branes which make angles of

2πn/12, where n = 0, ..., 5. One can check that for n even there are 3f equally spaced

ones and for n odd there are f equally spaced ones. In both special cases (τ = i and

τ = exp(2πi/3)) the total number of D1-branes is 12f .

We have thus seen that the classification of Dp branes for p > 0 is much more sporadic

than those for the D0 branes. This is to be expected because the D0 branes are precisely

the ones naturally picked out by the diagonal modular invariant.

29

7. RCFT and Higher Dimensional Calabi-Yau

So far we have talked about CFT’s based on the simplest Calabi-Yau sigma model,

namely the target being T 2. More precisely, we focused only on the bosonic sigma model,

but in this case the incorporation of fermions does not modify our discussion, as the

fermion partition function is independent of T 2 moduli. It is natural then to raise the

question of RCFT’s corresponding to supersymmetric sigma models propagating on higher

dimensional Calabi-Yau manifolds.

Unfortunately not much is known about the exact solutions in such cases, and we

only have existence proof for such CFT’s. The only general classes we know are tensor

products of minimal N = 2 supersymmetric conformal theories (Gepner models) and

toroidal orbifolds. It turns out that both classes are RCFT’s or have moduli for which

RCFT’s appear in a dense subspace. It is the existence of this class of examples which

motivates the belief that CFT’s are “exactly solvable” if they are rational or near one.

Consider quintic threefold, for example. There is only one point in its moduli of Kahler and

complex deformation where the CFT is exactly known and that is the RCFT corresponding

to the Gepner point. One wonders whether there are other points on the moduli space

where they are rational and therefore, perhaps solvable.

In this section we propose a criterion for rationality of conformal theory on Calabi-Yau

manifolds which agrees with all the known examples of rational points discussed above (see

[35] for a further evidence in the case of toroidal conformal field theories). However, given

mathematical conjectures a la Andre and Oort [36,37] our proposal for rationality suggests

that RCFT’s are not dense in the generic case of Calabi-Yau sigma models!

Our criterion for rationality is motivated by generalization of the notion of complex

multiplication to higher dimensional varieties. Indeed, it was pointed out to us by Kazhdan

and Mazur that there already exists a suitable notion of complex multiplication5 for higher

dimensional varieties introduced in 1969 by D. Mumford [8]. In particular, complex mul-

tiplication was studied in the context of K3 surface by Piateckii-Shapiro and Shafarevich

[9], and more generally, in the context of Calabi-Yau manifolds by Borcea [10]. The idea

is rather simple: One first defines what it means for an abelian variety (i.e. complex tori)

to admit complex multiplication. Then one asks if the variation of the Hodge structure of

the Calabi-Yau M and its mirror W , whose period matrices lead to a pair of associated

abelian varieties admit complex multiplication (of a ‘compatible’ type).

5 Mathematically, it says that manifold M has complex multiplication when its Hodge group,

Hg(M), is commutative.

30

7.1. Complex Multiplication for Complex Tori

Consider a complex n-dimensional torus.

T 2n ∼=Cn/ZZ2n

This is defined by identifications

zi ∼ zi + δij zi ∼ zi + Tij

where T is an n×n complex symmetric period matrix. Then we say that the torus admits

complex multiplication if there exists a non-trivial endomorphism

z 7→ Az (7.1)

which implies that

A =M +NT

T A =M ′ +N ′T

for some integer matrices M,N,M ′, N ′. In other words we have a second order matrix

equation

T (M +NT ) =M ′ +N ′T ⇒

T NT + TM −N ′T −M ′ = 0. (7.2)

Moreover one requires that N has rank n.6

7.2. Calabi-Yau and the Intermediate Jacobian

The notion of considering mid dimensional cohomology elements and integrating over

a mid dimensional integral lattice of cycles to define an abelian variety is well known. For

example for the case of a genus g Riemann surface with a symplectic pairing of 1-cycles:

(Ai, Bj) = δij (Ai, Aj) = (Bi, Bj) = 0 (7.3)

one considers g holomorphic 1-forms ωi normalized relative to the A-cycles∫

Aj

ωi = δij

6 This rules out examples like M = ECM × E′, where ECM is an elliptic curve with complex

multiplication and E′ is another arbitrary elliptic curve without CM.

31

and defines the Jacobian by ∫

Bj

ωi = Tij . (7.4)

Similar idea works for arbitrary complex varieties and in particular for Calabi-Yau 3-folds.

In the case of Calabi-Yau threefold Tij can be identified with the complex torus defining the

coupling constants of the associated U(1)n gauge fields and is related to the prepotential

F (in the homogeneous coordinates) by

Tij = ∂i∂jFWe say that the Calabi-Yau admits complex multiplication if the corresponding interme-

diate Jacobian associated with T admits complex multiplication.

7.3. A Criterion for RCFT for Calabi-Yau Sigma Models

A Calabi-Yau sigma model is completely characterized by its complex and Kahler

moduli. Since the notion of complex multiplication is natural only for complex structure

of the variety, it is natural to associate to a given Calabi-YauM with a given complex and

Kahler structure, a mirror pair of Calabi-Yau (M,W ) with fixed complex structures (where

we have traded the Kahler structure of the original Calabi-Yau with complex moduli of

its mirror). This is effectively how we studied the case of elliptic curve, by viewing τ and

ρ as defining pairs of elliptic curves. We propose the following criterion of the Calabi-Yau

sigma model to correspond to a RCFT:

Sigma model on Calabi-Yau corresponding to the pair (M,W ) is RCFT if and only if

M and W admit complex multiplication over the same number field.

For example, for a Calabi-Yau threefold M satisfying complex multiplication one gets

an equation (7.2) of order two in the (h2,1(M)+1)× (h2,1(M)+1) matrix T . In this case,

it has been shown by Borcea [10] that existence of complex multiplication is equivalent to

the condition that elements of the endomorphism matrix A generate imaginary number

field K:

K ∼= End(H3(M,Q))⊗Q (7.5)

of degree:

[K : Q] = 2(h2,1(M) + 1)

On the other hand, the mirror W admitting complex multiplication gives elements which

are in an algebraic number field of degree 2(h2,1(W )+1) = 2(h1,1(M)+1), and the criterion

we are imposing for RCFT is that they are elements of the same number field7.

7 Clearly, the degree of this number field is bounded by min(2(h2,1(M) + 1), 2(h1,1(M) + 1)).

32

7.4. Application of the Criterion

To check the criterion, we have to make sure it agrees with the known cases of RCFT’s

for Calabi-Yau sigma models. Indeed it does. Toroidal orbifolds corresponding to RCFT’s

obviously admit complex multiplication inherited from the fact that the underlying torus

admits complex multiplication (extending our discussion from the elliptic case — the sim-

plest case being orbifolds of the product of elliptic curves). Much more non-trivial are the

Gepner points, corresponding to Fermat polynomials. It is also known that these also do

admit complex multiplication [38,39,40,10]. Below we show how this works for the quintic

threefold with one complex moduli. This is already impressive evidence for the criterion

we have proposed for rationality.

We now wish to study how frequently one would encounter rational conformal theory

in the moduli of a given Calabi-Yau sigma model, assuming the criterion we have proposed

holds. To get a feel for this, consider Riemann surfaces. As discussed above we can identify

with it an associated Jacobian. However the moduli space of genus g curves is 3g − 3

complex dimensional whereas the moduli space of the abelian varieties of dimension g has

dimension g(g+1)/2. Thus for g > 4 the Riemann surfaces are not dense in moduli of the

corresponding tori. The Schottky problem is to identify which abelian varieties can arise

for Riemann surfaces.

Similarly, one could ask which Riemann surfaces admit complex multiplication. Even

though there is a dense set of point in the moduli of complex structure of the tori ad-

mitting complex multiplication this may not hold true for the measure zero subspace of

it corresponding to those coming from Riemann surfaces. Unlikely as this sounds, indeed

there is evidence and a standing mathematical conjecture by Coleman [41] (see also [42] for

recent developments) that for sufficiently large g there are only a finite number of Riemann

surfaces admitting CM! Indeed a similar conjecture exists for arbitrary varieties8 and it is

believed that the number of CM points are dense only if the relevant moduli space itself

is of the form G/H (i.e. a submoduli of the full toroidal moduli defined by some linear

8 Mathematically, a basic version of this conjecture is known as Andre-Oort conjecture [36,37],

and we thank F. Oort and B. Mazur for explaining to us the general philosophy behind it. Roughly,

Andre-Oort conjecture says that in order for a (sub)family of algebraic varieties to contain a dense

set of CM-points, the corresponding moduli space has to be “Shimura (sub)variety”. For example,

the moduli spaces of elliptic curves and K3 surfaces are of this type, however the moduli space of

a Calabi-Yau manifold in general is not.

33

algebraic constraint). In particular, in the case of complex tori (of complex dimension n)

and K3 surfaces this conjecture predicts dense set of CM/RCFT points. Indeed, in both

cases the moduli space turns out to be a coset space:

SO(2n, 2n;ZZ)\SO(2n, 2n)/SO(2n)× SO(2n) (7.6)

and

SO(20, 4;ZZ)\SO(20, 4)/SO(20)× SO(4)

respectively. On the other hand, for the case of the one parameter family of quintic three-

folds, complex multiplication is conjectured to occur at most at finite number of points.

It would be very interesting to test this conjecture as it seems to be at odds with the

common lore for RCFT’s. This of course might be a blessing in disguise as it seems to

point to the existence of some finite number of interesting points on the moduli of Calabi-

Yau compactifications. These may end up being interesting points when the moduli of

Calabi-Yau manifolds get frozen by some mechanism.

7.5. The Example of Fermat Quintic

Finally, a non-trivial test of our criterion can be obtained by considering a one-

dimensional family of quintic three-folds

M : z51 + z52 + z53 + z54 + z55 − 5ψz1z2z3z4z5 = 0

At the Fermat point, ψ = 0, the corresponding sigma-model becomes rational, namely it

is the (k = 3)5 Gepner model. On the other hand, Calabi-Yau manifold M has complex

multiplication at ψ = 0 [10,38,39]. This is related to the fact that the automorphism

group is bigger for the Fermat quintic than for any other generic member in this family.

Moreover, ψ = 0 is the only known non-trivial CM-point in the whole moduli space of

M . In this sense, there is the same amount of physical and mathematical data on this

question, which therefore provides at least one non-trivial check of our proposal.

In order to see explicitly that the Fermat quintic has sufficiently many holomorphic

endomorphisms (and, therefore, admits complex multiplication) let us evaluate the period

matrix (7.4) at ψ = 0. In a particular basis of A and B cycles (7.3), the standard calculation

gives [13,43]:

T =

(α− 1 α+ α3

α+ α3 −α4

)

34

where α is a (non-trivial) 5-th root of unity, α5 = 1. Note, that α is a solution to the

degree 4 polynomial with integer coefficients:

x4 + x3 + x2 + x+ 1 = 0

It is straightforward to check that the matrix T satisfies the quadratic matrix equation

of the form (7.2):

T NT + TM −N ′T −M ′ = 0

where

N =

(1 −10 1

), M =

(0 00 0

), N ′ =

(−1 0−1 0

), M ′ =

(−1 0−1 −1

),

The corresponding endomorphism is given by the matrix:

A =

(α− 1 α + α3

1 + α+ α3 −α4

)

Notice, that elements of T and A take values in a degree 4 number field K, cf. (7.5):

K =Q(α)

which can be obtained from the field of rational numbers, Q, by adjoining the fifth root of

unity. Since in the present example h2,1(M) = 1, this is in complete agreement with the

general formula for the degree, [K : Q] = 2(h2,1 + 1).

Acknowledgments

We would like to thank D. Kazhdan and B. Mazur for many illuminating discussions

on complex multiplication. We are also grateful to J. de Jong, J. Maldacena, K. Oguiso,

H. Ooguri, F. Oort, A. Recknagel, S. Shenker, F. Rodriguez-Villegas, and E. Witten for

valuable discussions. This research was partially conducted during the period S.G. served

as a Clay Mathematics Institute Long-Term Prize Fellow. The work of S.G. is also sup-

ported in part by grant RFBR No. 01-02-17488, and the Russian President’s grant No.

00-15-99296. The work of C.V. is supported in part by NSF grants PHY-9802709 and

DMS 0074329.

35

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